New weight-shifting operators now follow
from the Cayley-Menger parametrisation (REF ). Acting on the Schwinger exponent (REF ) with an appropriate polynomial differential operator in the momenta pulls down a corresponding polynomial in the \(s_{ij}\) .
Choosing these polynomials to be the Cayley-Menger determinant and its minors, we obtain shift operators either increasing \(\alpha \)  or decreasing one of the \(\alpha _{ij}\)  by integer units.
We discuss these new operators in section REF , showing their effect is to increase the spacetime dimension by two while performing assorted shifts of the operator dimensions.
Further weight-shifting operators can then be constructed by conjugating these operators with shadow transforms as shown in section REF .
Explicit examples are given for the 3- and 4-point functions in section REF .
Then, in section REF , we turn to analyse the weight-shifting operators proposed in [1]}. These preserve the spacetime dimension but their action can nevertheless be understood using our parametric representations.
